Mixed-mode Fracture Mechanics Research Articles

Mixed mode I/II fracture behavior and surface morphology of hard rock under dynamic loading

To investigate the dynamic mixed mode I/II fracture properties of hard rock, split-Hopkinson pressure bar (SHPB) was utilized to conduct dynamic fracture tests on cracked straight through Brazilian disc (CSTBD) diorite specimens with different mode mixities. The dynamic fracture load and fracture toughness for different crack angles or mode mixities were determined. The dynamic fracture process and crack mouth opening displacement (CMOD) of CSTBD specimens were monitored by high-speed camera and digital image correlation (DIC) technique, and the fractal and morphology characteristics of fracture surfaces were further analyzed using laser 3D scanner and scanning electron microscope (SEM). The test results indicate that the dynamic CMOD behavior is interrelated closely to the force acting on the crack surface, and the allowable CMOD corresponding to the dynamic peak load decreases with increasing crack angle. In addition, the stress-based and strain-based fracture criteria were introduced to analyze the mixed mode fracture mechanism of diorite. It is found that the GMTS criterion with small fracture process zone (FPZ) size and the EMTSN criterion have the best prediction effect for fracture toughness and crack initiation angle respectively. Observation results show that the macro fracture trajectories of primary cracks and secondary cracks are controlled by the prefabricated crack angle, while the variation in mode mixity has little impact on the meso-roughness and micro-morphology of the fracture surface. The fracture surface of CSTBD diorite specimen under dynamic load exhibits the SEM morphology features of mixed tensile-shear cracks.

A coupled mix-mode cohesive law based on a cylindrical potential function

A novel mixed-mode cohesive law derived from a potential function is presented. The potential function is formulated using physical parameters that can be extracted from any fracture mechanics test capable of providing R-curves in terms of the J-integral as a function of the normal and tangential end-openings. The proposed cohesive law is able to describe the fracture behaviour of composites with large fracture process zones, including fibre bridging. As such, it is capable of describing both the crack tip, as well as the bridging region. An important aspect of the formulation is that the shape of the mixed-mode cohesive laws are derived and not assumed. The mixed-mode cohesive law was tested using synthetic data emulating mixed-mode fracture mechanics tests. The cohesive tractions extracted from the method exhibited characteristics which were not seeded on the model such as negative normal tractions under pure mode shear loading and non-zero shear loading under pure normal mode loading.

Finite element analysis of fracture mechanics in micromorphic theory

In this work, the kinematics and basic laws of micromorphic theory are briefly introduced. Then, constitutive equations for micromorphic thermo-visco-elastoplastic solids are rigorously derived. Finite element equations for the displacement, micromotion and temperature fields are formulated based on the principle of virtual work. As examples, three-dimensional, dynamic, finite strain, mixed mode fracture mechanics problems have been solved by a general purpose in-house developed code. Numerical results, including temperature, [Formula: see text] norms of stresses, plastic strains and micromotions, are presented and the physical meanings are discussed. This paper provides theoretical fundamentals to study the thermo-visco-elastic-plastic behaviors of micromorphic solids and constructs a numerical framework to study fracture mechanics of edge-crack problems.

An anisotropic phase-field approach accounting for mixed fracture modes in wood structures within the Representative Crack Element framework

Wood exhibits complex behavior since the elasticity and the fracture evolution are characterized by strongly anisotropic features. Furthermore, the fracture resistance of wood is characterized as mode-dependent. Therefore, meaningful and promising model development for wood or timber structures is always challenging. The work at hand incorporates a mixed-mode fracture mechanism into an anisotropic phase-field approach for modeling wood failure. Therewith, distinctive fracture resistance and the phase-field driving energy are considered by different load modes. Furthermore, the aforementioned anisotropic and mode-dependent phase-field model is constituted within the conceptual Representative Crack Element (RCE) framework, which allows modeling of physical and correct crack deformations, including crack surface opening, closing, shearing, and mixed mode deformations. In particular, the nature of the RCE framework provides an explicit distinction between Mode I and Mode II behaviors. The governing equations are consistently derived and the formulations are implemented into the Finite Element Method context. Representative and demonstrative numerical studies have shown good agreement versus the corresponding experimental evidence. Several meaningful findings and potential perspectives are discussed to close this paper.

Theory and calculation of the mixed-mode fracture for coupled chemo-mechanical fracture mechanics

In this paper, a J-integral formula and numerical calculation method are presented, which confirm the path-independence of a J-integral, considering the mixed-mode fracture in a chemo-mechanical coupled medium. The Jk-integral denotes a two-dimensional vector consisting of the components J1 and J2. Using the Equivalent Domain Integral (EDI) method, the chemo-mechanical coupled J2-integral, with an equivalent domain form that includes both a surface integral and a line integral, is deduced using a poroelastical model. Based on a semi-analytical method, an auxiliary J¯2-integral is derived to develop a finite element calculation program. By choosing two characteristic lengths, near the crack tip, the mixed-mode fracture parameters, KI, KII and Ts (T stress) at the crack tip, can be calculated. The mixed-mode fracture problems, which are related to center-cracked and edge-cracked plates, are numerically studied for certain chemo-mechanical coupled boundary conditions. The accuracy of this method is verified with numerical examples, which shows the path-independence of the J-integral for chemo-mechanical coupling. The stress fields of two different cracked hydrogel plates are analyzed with different boundary conditions under the chemo-mechanical coupled mixed-mode fracture. This work provides new insights into the mixed-mode fracture mechanics of a chemo-mechanical coupled medium.

Flexible film separation analysis of LCD based mask stereolithography

Mask stereolithography technology uses ultraviolet masks to print curable liquid materials. A key process for layer-by-layer bottom-up mask printing is the separation of cured layer from the vat bottom. For film-assisted separation, most studies focused on the separation process of the inert film that bonded to the vat bottom. The separation characteristics and mechanism of the widely used flexible film has not received much attention. In this investigation, based on liquid crystal display (LCD) mask stereolithography, separation processes of the inert film in both semi-flexible and flexible states were compared and analyzed. The results showed that the average maximum separation force of flexible film was reduced above 48 %, and the average separation energy was reduced above 22 % compared with the semi-flexible film for both large-area and array supports models. The flexible film can smooth separation process by reducing the stiffness of separation force curve. The reducing effect of separation force of flexible film was achieved by its large deformation and mixed-mode fracture mechanism.

Fracture Mechanics Assessment of Large Diameter Wind Turbine Bearings

The structural integrity of large diameter wind turbine bearings have been investigated using the built-in “contour integral” tool in ABAQUS finite element software package by modeling three-dimensional penny-shaped cracks and evaluating the stress intensity factors. In order to sub-model a crack and investigate the fracture mechanics of the rolling contact between the rollers and the raceway, a python script was developed and implemented in the analysis. Important steps to build the crack model are detailed and recommendations are made for further use of the finite element modeling tool in compressive mixed mode fracture mechanics assessment of wind turbine bearings. Moreover, the influence of initial residual stresses due to induction hardening of the raceway is also investigated and discussed in this paper. For frictionless contacts between the two crack faces, “contour integral” in ABAQUS appears to be a suitable method to obtain accurate stress intensity factor solutions for modes II and III. The results from this study are validated through comparison with the analytical solutions available in the literature and are expected to facilitate numerical life assessment of wind turbine bearings.

On mixed-mode fracture mechanics models for contact area reduction under shear load in soft materials

The fundamental problem of friction in the presence of macroscopic adhesion, as in soft bodies, is receiving interest from many experimentalists. Since the first fracture mechanics ‘purely brittle’ model of Savkoor and Briggs, models have been proposed where the mixed mode toughness is interpreted with phenomenological fitting coefficients introducing weaker coupling between modes than expected by the “purely brittle” model. We compare here two such previously proposed models and introduce a third one to show that the transition to sliding is very sensitive to the form of the mixed-mode model. In particular, after a quadratic decay of the contact area with load for modest tangential loads, depending on the exact form of the mixed mode function, there is an inflexion point and an asymptotic limit, or a jump to the Hertzian contact area. We find also that the unstable points are different under load or displacement control. Hence, the form of the mixed mode function, and not only its parameter, is an extremely sensitive choice.

Mixed Mode Fracture Mechanics Behaviour of PMMA

SummaryThe mixed mode brittle fracture (mode I/mode II) of poly(methyl methacrylate) (PMMA) was investigated using single‐edge notched tensile (SENT) specimens by applying a mixed mode special loading device which allows varying the loading angle from 0° (pure mode I) to 90° (pure mode II). The experimental results dealing with mixed mode fracture toughness and initial angle of crack propagation are in good agreement with predictions based on a modified criterion of maximum tangential stress (MTS criterion) where non‐singular stresses (T stresses) are considered. These additional T stresses are the reason why distinct fracture by normal stress with crazing is always macroscopically observed in PMMA under the selected loading conditions including pure external mode II loading. Therefore, the length of the craze zone, i.e. the process zone, has been determined to be about 0.5 mm independent of the loading angle. It has to be pointed out that the fracture surface morphology should be kept in mind more often in future for assessing the mixed mode fracture mechanics parameters.

J-integral evaluation for 2D mixed-mode crack problems employing a meshfree stabilized conforming nodal integration method

Two-dimensional (2D) in-plane mixed-mode fracture mechanics problems are analyzed employing an efficient meshfree Galerkin method based on stabilized conforming nodal integration (SCNI). In this setting, the reproducing kernel function as meshfree interpolant is taken, while employing the SCNI for numerical integration of stiffness matrix in the Galerkin formulation. The strain components are smoothed and stabilized employing Gauss divergence theorem. The path-independent integral (J-integral) is solved based on the nodal integration by summing the smoothed physical quantities and the segments of the contour integrals. In addition, mixed-mode stress intensity factors (SIFs) are extracted from the J-integral by decomposing the displacement and stress fields into symmetric and antisymmetric parts. The advantages and features of the present formulation and discretization in evaluation of the J-integral of in-plane 2D fracture problems are demonstrated through several representative numerical examples. The mixed-mode SIFs are evaluated and compared with reference solutions. The obtained results reveal high accuracy and good performance of the proposed meshfree method in the analysis of 2D fracture problems.

Effect of Joint Scale and Processing on the Fracture of Sn-3Ag-0.5Cu Solder Joints: Application to Micro-bumps in 3D Packages

In 3-dimensional (3D) packages, a stack of dies is vertically connected to each other using through-silicon vias and very thin solder micro-bumps. The thinness of the micro-bumps results in joints with a very high volumetric proportion of intermetallic compounds (IMCs), rendering them much more brittle compared to conventional joints. Because of this, the reliability of micro-bumps, and the dependence thereof on the proportion of IMC in the joint, is of substantial concern. In this paper, the growth kinetics of IMCs in thin Sn-3Ag-0.5Cu joints attached to Cu substrates were analyzed, and empirical kinetic laws for the growth of Cu6Sn5 and Cu3Sn in thin joints were obtained. Modified compact mixed mode fracture mechanics samples, with adhesive solder joints between massive Cu substrates, having similar thickness and IMC content as actual micro-bumps, were produced. The effects of IMC proportion and strain rate on fracture toughness and mechanisms were investigated. It was found that the fracture toughness GC decreased with decreasing joint thickness (hJoint). In addition, the fracture toughness decreased with increasing strain rate. Aging also promoted alternation of the crack path between the two joint–substrate interfaces, possibly proffering a mechanism to enhance fracture toughness.

Dynamic Crack Growth Normal to an Interface in Bi-Layered Materials: An Experimental Study Using Digital Gradient Sensing Technique

The dynamic fracture behavior of layered architectures is experimentally studied. Specifically, crack penetration, trapping, and branching at an interface are examined. A newly introduced optical technique called Digital Gradient Sensing (DGS) that quantifies elasto-optic effects due to a non-uniform state of stress is extended to perform full-field measurements during the fracture event using ultrahigh-speed photography. By exploiting the richness of two simultaneously measured orthogonal stress gradient fields, a modified approach for extracting stress intensity factors (SIFs) is implemented for propagating crack-tips under mixed-mode conditions. The method is first calibrated using a quasi-static experiment complemented by finite element simulations before implementing it for studying dynamic mixed-mode fracture mechanics of layered configurations. The layered systems considered consist of two PMMA sheets bonded using an acrylic adhesive with the interface oriented normally to the initial crack propagation direction. Interfaces are characterized as ‘strong’ and ‘weak’ by their crack initiation toughness. The dynamic fracture of monolithic PMMA sheet is also studied in the same configuration for comparison. The crack growth and fracture parameter histories of propagating cracks are evaluated. The interface is shown to drastically perturb crack growth behavior resulting in higher dissipation of fracture energy by exciting crack trapping, branching, and mixed-mode growth mechanisms.

Graphical User Interface (GUI) for Mixed Mode Fracture Mechanics Analysis of Plates Using Crack Extension Technique

In this paper, a new approach for the mixed mode fracture mechanics analysis of plates using crack extension technique, by developing a new graphical user interface (GUI), has been introduced. The main objective of the paper is to show the usefulness and power of Matlab GUI in investigating and analyzing the effects of crack configurations and orientations on the calculation of stress intensity factors for cracked plates. The second purpose of this paper is to carry out an analysis using the developed GUI to simulate different crack configurations such as central-crack plate, single-edge plate, and double-edge plate. Illustrated problems in the field of analysis of fracture mechanics are carried out. The results show that the developed GUI is very useful for engineers, designers, and analysts of fracture mechanics problems.

The mixed-mode fracture mechanics analysis of an embedded arbitrary oriented crack in a two-dimensional functionally graded material plate

Mixed-mode fracture mechanics analysis of an embedded arbitrarily oriented crack in a two-dimensional functionally graded material using plane elasticity theory is considered. The material properties are assumed to vary exponentially in two planar directions. Then, employing Fourier integral transforms with singular integral equation technique, the problem is solved. The stress intensity factors (SIFs) at the crack tips are calculated under in-plane mechanical loads. Finally, the effects of crack orientation, material non-homogeneity, and other parameters are discussed on the value of SIF in mode I and mode II fracture.

On Crack Propagation of I-II Mixed Mode Fracture in Concrete by Means of Photoelastic Coatings

Investigation on crack propagation is important to understand mechanical behaviors and fracture mechanism in concrete. Till now substantial work has been done on crack propagation under mode-I loading, and mixed-mode fracture mechanics problems, however, are more common for actual crack growth. This study aims to systematically investigate the crack initiation, stable propagation and unstable failure of concrete under I-II mixed mode condition by using photoelasticity technique. For this purpose, reflective photoelastic coatings were applied to the surface of the four-point shearing notched concrete specimens with the maximum size of 2500mm×600mm×200mm. The resulting interference fringe on photoelastic coatings was recorded to reflect a complete crack propagation process. Based on the colored strain sequence, the strain profile around the crack tip could be determined and thereafter the load-crack extension curve. Then finite element analysis was performed to simulate the process of crack propagation and corresponding stress intensity factors were solved for every step. Experimental results show obvious stable crack propagation in concrete before unstable failure under I-II loading condition. The conventional failure criterion with the absence of this stable propagation quantity is therefore conservative. Besides, the fracture mode is ultimately I-mode even the specimen is subjected to I-II mixed mode loading condition, that is, the crack will extend in a direction perpendicular to the peak principal stress.

Role of microtexture in the interaction and coalescence of hydrogen-induced cracks

The role of microtexture in hydrogen-induced crack interaction and coalescence is investigated in line pipe steels using electron backscatter diffraction. Experimental evidence shows that, depending on the local grain orientation, crack interaction and coalescence can depart from the conditions predicted by the mixed-mode fracture mechanics of isotropic linear elastic materials. Stress simulation and microtexture analysis are used to explain the experimental observations.

734 回転自由度を有する四面体要素用の仮想き裂閉口積分法による混合モード破壊力学解析(OS20.破壊力学とき裂の解析・き裂進展シミュレーション(3),オーガナイズドセッション)

734 回転自由度を有する四面体要素用の仮想き裂閉口積分法による混合モード破壊力学解析(OS20.破壊力学とき裂の解析・き裂進展シミュレーション(3),オーガナイズドセッション)

DCB-specimen loaded with uneven bending moments

A double cantilever beam specimen loaded with uneven bending moments (DCB-UBM) is proposed for mixed mode fracture mechanics characterisation of adhesive joints, laminates and multilayers. A linear elastic fracture mechanics analysis gives the energy release rate and mode mixity analytically for both isotropic and orthotropic materials. By varying the ratio between the two applied moments, the crack tip stress state can be varied from pure mode I to pure mode II for the same specimen geometry. The specimen allows stable crack growth. A special test fixture is developed to create uneven bending moments. As a preliminary example, the DCB-UBM specimen was used for characterising fracture of adhesive joints between two laminates of thermoset glass fibre reinforced plastic.

712 テトラ二次要素用三次元仮想き裂閉口積分法(VCCM)による混合モードき裂解析(OS13(3) エネルギー機器構造物の劣化現象の理解と健全性評価)

712 テトラ二次要素用三次元仮想き裂閉口積分法(VCCM)による混合モードき裂解析(OS13(3) エネルギー機器構造物の劣化現象の理解と健全性評価)

Mixed mode fracture toughness of lead–tin and tin–silver solder joints with nickel-plated substrate

This study uses modified compact tension specimens to evaluate the fracture toughness of solder joints under a combination of tensile and shear loads. Copper substrates were coated with electroless nickel plating before they were soldered together. The samples were reflowed to peak temperatures of 215 and 240 °C for Sn–37Pb and Sn–3.5Ag, respectively, with appropriate preheating temperatures, holding time and cooling rates. Fracture toughness tests were conducted as per ASTM E399-90. Cross-sectional microstructural and fractography studies were carried out on the tested samples. Earlier fracture studies which were based on J integral as per ASTM E813-89 did not produce valid results because of brittleness of the solder joints. The current results showed that the fracture of the solder joints follows the general principles of a mixed mode fracture mechanism map and shear is the preferred mode of failure regardless of different mode of loadings, type of solders and substrate coatings.